Convex Hypersurfaces and $L^p$ Estimates for Schrödinger Equations

Details Convex Hypersurfaces and $L^p$ Estimates for Schrödinger Equations Convex Hypersurfaces and $L^p$ Estimates for Schrödinger Equations

2004-03-19

arxiv.org/abs/math/0403321v2

Mathematics

Analysis of PDEs

18 pages

Scientific paper

This paper is concerned with Schr\"odinger equations whose principal operators are homogeneous elliptic. When the corresponding level hypersurface is convex, we show the $L^p$-$L^q$ estimate of solution operator in free case. This estimate, combining with the results of fractionally integrated groups, allows us to further obtain the $L^p$ estimate of solutions for the initial data belonging to a dense subset of $L^p$ in the case of integrable potentials.

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